Äîêë. Àêàä. Íàóê ÑÑÑÐ
Òîì 281 (1985), N2 4
LOWER
Soviet Math. Dokl.
Vol. 31 (1985), No.2
BOUNDS
FOR ÒÍÅ
MONOTONE
COMPLEXITY
OF SOME
BOOLEAN
FUNCTIONS
UDC 519.95
À. À. RAZBOROV
The combinatorial
complexity
L f î! à Boolean !unction
!(Õl,
...,
Õï) iB the least ïèò.
ber of logical elementB AND, OR and NOT neceBBary for itB realization
in the form of
à functional
Bcheme. It iB well known (âåå, for example,
[1]) that there are Boolean
functionB whoBe combinatorial
complexity
iB an exponential
function
of the number of
variableB. In à recent article [2], à natural Bequence of Boolean functionB
(1)
!1(Õ1,...,Õï1)'!2(Õ1,...,Õï2)'...'!ò(Õ1,...,Õïò)'...
was conBtructed, with Lfm ?: ñïò,
where ñ > 1 iB à univerBal conBtant.
In thiB note we will reBtrict ourBelveB to the conBideration
of BequenceB of the form (1)
BatiBfying the following condition:
the language {(ål ...enm)lm
Å N, !ò(å1, ...,Å.ïò) =
1} in the alphabet
{0,1} can Üå recognized Üó à nondeterminiBtic
Thring machine in
time which iB polynomial
in the length of the input ïò (i.e. it iB an N P-language). SUCh
BequenceB will Üå called con8tructive.
It iB intereBting to obtain lower boundB on the combinatorial
complexity
of funCtions
from the conBtructive Bequence (1), for example, in connection with the fol1owing remark
(derivable from the reBultB of [3]): ifthere iB à conBtructive
Bequence ofthe form (1) sUCh
that
-:1
lm
ò--îî
then Ð i= N Ð .Apparently
where an example
The
monotone
= 00,
the BtrongeBt reBult obtained
of à conBtructive
complexity
logLfm
log ïò
Lt
in thiB direction
Bequence (1) iB conBtructed
î! à monotone
Boolean
with
!unction
iB found in [41,
Lfm ?: 2.5ïò.
!(Õl,
...,Õï)
is the
least number of functional
elementB OR and AND neceBBary for itB realization
in the
form of à functional
Bcheme (without
the element NOT). Clearly Lt ?: Lf, and therefore
the problem
monotone
of finding
Boolean
was obtained
asymptotic
functionB
lower boundB on Lt
iB Bimpler.
for conBtructive
The beBt bound
Bequences (1) of
of thiB type known
until llOW
in [5] :
.L f+
m
2
,
?: ñ -} ïò
ognm
ñ > 0,
for à certain conBtructive
Bequence of the form (1).
In thiB note we Bhall conBtruct two conBtructive
BequenceB of monotone
tionB for which
LtM
?: n~lognm),
with
ñ
> 0.
The general
reBult
from
Boolean func.
which these
boundB òàó Üå obtained iB Btated in Theorem
1. TheoremB 2 and 3 are devoted to
boundB for the monotone complexity
of functionB from Bpecific conBtructive
BequenCes.
In order to formulate the reBults, it iB convenient to interpret
à Boolean function as the
Bet of inputB on which it takeB the value 1.
More preciBely, let R = { ål, ..., åï} Üå à finite Bet, and Âï = P(R) itB power set.
We define à bijection õ: Âï -+ {0,1}ï
in the fol1owing way: for Å Å Âï we Bet õ(Å) =
(ål, ...,
åï),
where ei = 0 if ei ~ Å, and ei = 1 if ei Å Å.
1980 Mathematic8SulrjectClassification(1985 Re1I.:9ion).Primary 65Q25.
@1985 American
0197-6788/85
354
Mathematical
Soci.ty
$1.00 + $.25 ð.' pag.
Òî the Boolean function f(Xl, ., ., Õï), of n variables we assign the set Àè) Å Ð(Âï)
in the following way: Àè) = {Å Å Bnlf(X(E))
= 1}. Clearly À gives à bijection
betweenthe set ofall Boolean functions ofn variables aIld Ð(Âï), for which Àè1& f2) =
A(h) n Àè2) and Àè1 v f2) = Àè1) U Àè2). We call the set ì Å Ð(Âï) monotone if
forall El, Å2 Å Âï, from Å1 Å ì and Å1 ~ Å2 it follows that Å2 Å Ì. We remark that
à Boolean function f is monotone if and only if the set Àè) i8 monotone. We denote
ÜóÐ+(Âï) the family of all monotone subset8 of Âï. Among the element8 of Ð+(Âï)
thereare, for example, the 8et8 À(O) = 0, À(l) = Âï, and A(Xi) = {Å Å Bnlei Å Å}.
Now 8èððîâå âîòå family ò of monotone 8ub8et8 of the 8et Âï i8 given; that is,
!JJt~ Ð+(Âï). We call ò à regular lattice if the following two condition8 are 8ati8fied:
;
à) {À(o), À(l), À(õl), ..., À(õï)} ~ ò.
Ü) If ò i8 regarded as à partiaIly ordered 8et under inclu8ion, them ò i8 à lattice
with re8pect to thi8 order .
The operation8 of taking greate8t lower and least upper bound8 wiIl Üå denoted Üó ï
andu re8pectively. We introduce the notation
á-(Ì1,
Ì2)
~ {Ì1
u M2)\(M1U
á+(Ì1,
Ì2)
~ (Ì1
n Ì2)\(Ì1
Ì2),
ï Ì2).
SuPPo8ethat we are given 8îòå monotone Boolean function f(Xl, ..., Õï) and à regularlattice ò. ÒÜå distance ðè, ò) between f and ò i8 defined to Üå the least natural
numbert for which there are element8 Ì, Mi and Ni of ò, i:::; i:::; t, 8èñÜ that
t
M~A(f)UUá-(Mi,Ni),
i=l
)
t
U u 6+(Mi, Ni):
i=l
It is relatively 8imple to prove the foIlowing
)
Àè)
~ ì
;,t(!
'Lj'i,-'
: THEOREM 1. For àïó monotone Boolean function f(Xl, ..., Õï) and àïó regular
f/attice !m ~ Ð+(Âï) the inequality Lj ~ ðè, ò) holds.
We now turn to the construction of con8tructive 8equences consi8ting of monotone
Booleanfunction8 of sufficiently great monotone complexity. The fir8t example corre8pondsto finite fragment8 of the N P-complete problem CLIQUE.
Let ò and s Üå natural numbers with s < ò, and let v = { Vl, ..., Vm} Üå à finite set.
Wesetn = ò(ò-1)/2
and R = {(Vi, vj)11 :::;i < j :::; ò} (the order in which the element8
ofR are indexed i8 irrelevant). For every W ~ v we define Ew Å Âï (Âï = P(R)) in
thefollowing way:
Ew ~ {(Vi,Vj) ERlvi,Vj Å W}.
Furthermore, we set
Ç(ò,Â) = {ÅÅ
Bnl3W (W ~ v & card W = s & Ew ~ Å)}.
3(ò, â) con8i8t8 of tho8e Å for which the graph (V, Å) contain8 à clique of 8ize at least
8. It i8 clear that Ç(ò,Â) i8 monotone. Suppo8e that Ðò,8(Õ1,...,õï)
= À-1(Ç(ò,â))
i8 the corresponding monotone Boolean function. À lower bound for Ljm.. i8 obtained
on the basi8 of Theorem 1 u8ing à certain regular lattice òò,8.
We wiIl de8cribe the
construction of òò,8 in general terms.
We introduce the foIlowing notation: !2{ = {WIW ~ v and card W :::; â- 1}; r ~!
[2se8lnò] .We define à binary relation S ~ !2{ õ !2{r in the foIlowing way:
(WO,(W1,...,Wr))ES
ifandonlyif
'v'i,j(l:::;i<j:::;r*WinWj~Wo).
355
The fact that (Wo, (W1, ..., Wr)) Å s will Üå more briefl.y expressed in the form W1, ...,
Wr f- WO.
Furthermore, if 1:2(1~ 1:2(2and W Å 1:2(,then the expression 1:2(1f- W signifies that
there are W 1, ..., W r Å 1:2(
1 with W 1, ..., W r f- W. À set 1:2(
1 ~ 1:2(
will Üå called clo8edif
\iW Å 1:2((1:2(1f- W => W Å 1:2(1).SinCe the intersection of closed sets is closed, there is à
smallest closed subset l:2(i ~ 1:2(
containing 1:2(1,
for àïó 1:2(1
~ 1:2(.
For à closed 1:2(1
~ 1:2(
we define the element rl:2(l' Å P+(Bn) in the following way:
rl:2(l' = {Å Å Bnl3W Å 1:2(1(Ew~ Å)}.
Finally, we set rotm,8 = {rl:2(l'1I:2(lclosed}.
LEMMA 1. à) rotm,8 i8 à regular lattice.
Ü) The lattice operation8 in rotm,8 have the !ollowing !îòò:
rQt1' ï rQt2' = Ã'Õ1n 'Õ2';
rQt1' u ã'Õ2' = Ã('Õ1u 'Õ2)"'.
The desired lattice rotm,8 has Üååï conStructed.
In estimating the quantity
Ð(!ò,8' rotm,8) from below, à key role is played Üó two lemmas stated below, which
we give without proof.
For an arbitrary 1:2(1
~ 1:2(
we denOte Üó 1:2(~
the subset of the minimal elements of ~1,
i.e.
1:2(~
= {W Å 1:2(11\iW'(W' ñ W => W' ~ 1:2(1)}.
LEMMA 2. I!l:2(l i8 clo8ed then card 1:2(~
~ (8 -1)!ò8-1.
Suppose that í = [8 -1]Ó is the set of funCtionS from v into {1, ...,8 -1}. For each
funCtion h Å Í, we define the ((8- l)-partite) graph Eh Å Bn Üó the equality
Eh = {(vi,vj)lh(Vi)
:f h(vj)}.
LEMMA 3. LetWo,W1,...,WrEl:2(andW1,...,Wrf-WO.
card{h Å HIEwo ~ Eh & Ew1 ~ Eh & ...&
Then
Ew. ~ Eh} ~ (1-
å-8)Ò .card Í.
From Lemmas 2 and 3 we obtain the following lower bound îï the distance.
LEMMA 4. p(fm,8'rotm,8)
:?:ffl8(83e81nm)-28.
From Lemma 4 and Theorem 1 the analogous bound for Ltm.. follows directly. In the
next theorem âîòå asymptotic properties of the bounds are established.
THEOREM 2. Âèððî8å that !ò,8(Õ1, ..., Xnm)' with ïò = ò(ò -1)/2,
i8 the òîïîtone Boolean !unction defined above, corre8ponding to the 8et î! tho8e graph8 îï ò vertice8 which contain à clique î! 8ize at lea8t 8. Then:
à) !îò 8 = conSt and ò-+ 00
Ltm.. :?: O(m8/(logm)28)j
Ü) !îò 8 = [ilnm]
and ò-+
00
L+fm.. :?: O (mClogm
),
ñ > Î.
REMARK 1. For comparison we mention the obvious upper bound
2
L+
< ~ .ò~
fm.. -2
8
)
.
The corresponding funCtional scheme in the elements AND and OR is easily conStructed
îï the basis of complete item-by-item examination of all elements of the set {WI
card W = 8}.
356
REMARK 2 .Both
of the sequences of Boolean functions considered in Theorem 2 are
constructive.
Our second example of à constructive sequence with the lower bound n![!lognm) on
its monotone complexity is à sequence of functions computing the logical permanent of à
Booleanmatrix. We specify à Boolean function !ò(Õ1,1, ..., Xi,j, ..., Õò,ò) of nm = ò2
variab1es
Üó the formUla
m
v
i~lXi,/1(i).
/1ÅÂò
!m(Xl,l"..,Xi,j,...,Xm,m)=
Wewill consider the graph-theoretical interpretation of this function.
We choose two disjoint sets of vertices v = {Vl,...'Vm}
and w = {Wl,...,Wm}.
Supposethat ei,j = (Vi,Wj) and R = {ei,jl1 :::; i,j :::; ò}. then Âï = P(R) turns out
to Üåexactly the set of all bipartite graphs with parts v and W, and Àèò) coincides
with the set of all bipartite graphs containing à perfect matching ( à per!ect matching in
à graph Å ~ v õ w is à set of ò edges having no vertices in common pairwise).
From the result of [61 the bound Lfm :::;O(ò5) follows for the combinatorial complexity. Îï the other hand, we have
THEOREM 3. Âèððîâå that !m(Xl,l,...,Xi,j,...,Xm,m)
is the logical permanent î!
àïò õ ò Boolean matrix. Then Ljm ~ mClogm, with ñ > Î.
The proof is similar in outline to the proof of Theorem 2 (the full proof of TheoremS
1and 3 will Üå published in an article in Mathematicheskie Zametki 37 (1985)).
Theorem 3 gives an affirmative answer to Pratt's question [71 as to whether the gap
betweenthe combinatorial and the monotone complexity of Boolean function can Üå
suprapolynomial in the number of variables.
ln conClusion I would like to thank À. L. Semenov, who interested òå in the subject
consideredhere, and S. I. Adyan, for his invaluable help in preparing this work for
publication.
Moscow State University
Received 14/JAN/85
BIBLIOGRAPHY
I. MichaelS. PaterBOn,JoumeesAlgorithmiques (Paris, 1975), Asterisque No. 38~39, Soc. Mat.
France,
Paris, 1976,ðð. 183-201.
2.N. Ê. Kosovskil, SeventhAII-Union Conf. Math. Logic, Abstracts î! Reports, Novosibirsk,1984,
ð.77.(Russian)
3.J. Å. Savage,J. Àââîñ. Comput. Mach. 19 (1972),660-674.
4. WolfgangJ. Paul, SIAM J. Computing 6 (1977),427--433.
5.IngoWegener,Theoret. Comput. Sci. 21 (1982),213-224.
6. JohnÅ. Hopcroft and Richard Ì. Êàãð, SIAM J. Computing 2 (1973),225-231.
7.VaughanR. Pratt, SIAM J. Computing 4 (1975),326-330.
TranslatedÜó G. L. CHERLIN
') ),j; '!) !
357